Wavelet packet computation of the Hurst exponent

Wavelet packet analysis was used to measure the global scaling behaviour of homogeneous fractal signals from the slope of decay for discrete wavelet coefficients belonging to the adapted wavelet best basis. A new scaling function for the size distribution correlation between wavelet coefficient energy magnitude and position in a sorted vector listing is described in terms of a power law to estimate the Hurst exponent. Profile irregularity and long-range correlations in self-affine systems can be identified and indexed with the Hurst exponent, and synthetic one-dimensional fractional Brownian motion (fBm) type profiles are used to illustrate and test the proposed wavelet packet expansion. We also demonstrate an initial application to a biological problem concerning the spatial distribution of local enzyme concentration in fungal colonies which can be modelled as a self-affine trace or an ‘enzyme walk’. The robustness of the wavelet approach applied to this stochastic system is presented, and comparison is made between the wavelet packet method and the root-mean-square roughness and second-moment approaches for both examples. The wavelet packet method to estimate the global Hurst exponent appears to have similar accuracy compared with other methods, but its main advantage is the extensive choice of available analysing wavelet filter functions for characterizing periodic and oscillatory signals.